I am reading a book on financial mathematics, and frequently encounter the phrase "predictable process", which I haven't seen definition of, and cannot find the definition online.

At first I thought that this was referring to a process which is known exactly at $t = 0$, but that is not the case, because then I see decomposition of an $(\mathcal F_t)_{t\in\mathbb N}$-adapted process $(X_t)$ into a martingale process $(M_t)$ and a predictable process $(A_t)$ where